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Nis28.12.2007P. P. FizievDepartment ofTheoretical PhysicsUniversity of Sofia


Exact Solutions of Regge-Wheelerand Teukolsky EquationsThe Regge-Wheeler (RW) equation describes the axial perturbations ofSchwarzschild metric in linear approximation.The Teukolsky Equations describe perturbations of Kerr metric.We present here:• Their exact solutions in terms of confluent Heun’s functions.• The basic properties of the RW general solution.• Novel analytical approach and numerical techniques for study ofdifferent boundary problems which correspond to quasi-normalmodes of black holes and other simple models of compact objects.• The exact solutions of RW equation in the Schwarzschild BH interior.• The exact solutions of Teukolsky master equations (TME).• New singular exact solutions of TME and their application to the theoryof the relativistic jets.


Linear perturbations ofSchwarzschild metric1957 Regge-Wheeler equation (RWE):The potential:The type of perturbations: S=2 - GW, s=1-vector, s=0 – scalar;The tortoise coordinate:The Schwarzschild radius:The area radius:1758 Lambert W(z) function: W exp(W) = z


The standard ansatzseparates variables.The “stationary” RWE:One needs proper boundary conditions (BC).Known Numerical studies andapproximate analytical methods for BH BC.See the wonderful reviews:V. Ferrary (1998),K. D. Kokkotas & B. G. Schmidt (1999),H-P. Nollert (1999).V. Ferrari, L. Gualtieri (2007).and some basic results in:S. Chandrasekhar & S. L. Detweiler (1975),E. W. Leaver (1985),N. Andersson (1992),and many others!


Exact mathematical treatment:PPF,In r variable RWE reads:The ansatz:reduces the RWE to a specific type of 1889 Heun equation:with


Thus one obtains a confluent Heun equation with:2 regular singular points: r=0 and r=1, and1 irregular singular point: in the complex planeNote that after all the horizon r=1 turns to be a singular pointin contrary to the widespread opinion.From geometrical point of view the horizon is indeeda regular point (or a 2D surface) in the SchwarzschildRiemannian space-time manifold:It is a singularity, which is placed in the (co) tangent fiberof the (co) tangent foliation:and is “invisible” from point of view of the base .


Different types of boundary problems:I. BH boundary problems: two-singular-points boundary.ҐUp to recently only the QNM problem on [1, ), i.e. on theBH exterior, was studied numerically and using differentanalytical approximations.We present here exact treatment of this problem, as well asof the problems on [0,1] (i.e. in BH interior), and on [0, ).


QNM on [0, ) by Maple 10:Using thecondition:-iOne obtains by Maple 10 for the first 5 eigenvalues:and 12 figures - for n=0:


Perturbations of the BH interiorMatzner (1980), PPF gr-qc/0603003,PPF JournalPhys. 66, 0120016, 2006.Forone introduces interior time:and interior radial variable: .Then:where:


The continuous spectrumNormal modes in Schwarzschild BH interior:A basis for Fourier expansion of perturbations of general formin the BH interior


The special solutions with :These:• form an orthogonal basis with respect to the weight:• do not depend on the variable .• are the only solutions, which are finite at both singular endsof the interval .


The discrete spectrum -pure imaginary eigenvalues:Ferrari-Mashhoon transformation: For :“falling at the centre” problemoperator with defect Additional parameter – mixing angle : Spectral condition – for arbitrary :


Numerical resultsFor the first 18 eigenvaluesone obtains:For alpha =0 – no outgoing waves:Two potential weels –> two series:Two series: n=0,…,6; andn=7,… exist. The eigenvaluesIn them are placed around thelinesand .


Perturbations of Kruskal-Szekeres manifoldIn this case the solution can be obtained from functionsimposing the additional condition which may create a spectrum:It annulates the coming from the space-infinity waves.The numerical study for the case l=s=2 shows that it is impossibleto fulfill the last condition and to have some nontrivial spectrum ofperturbations in Kruskal-Szekeres manifold.


II. Regular Singular-two-pointBoundary Problems atPPF,Dirichlet boundaryCondition at :The solution:Physical meaning:Total reflectionof the waves atthe surface witharea radius :The simplest model of a compact object


The Spectralcondition:Numerical results: The trajectories in ofThe trajectory of the basiceigenvalueinand the BH QNM (black dots):


The Kerr (1963) MetricIn Boyer - Lindquist (1967) - {+,-,-,-} coordinates:


The Kerr solution yields much more complicatedstructures then the Schwarzschild one:The event horizon, the Cauchy horizonand the ring singularityThe event horizon, the ergosphere,the Cauchy horizon andthe ring singularity


Simple algebraic and differential invariantsfor the Kerr solution:Let is the Weyl tensor, - its dualLet- Density forthe Eulercharacteristicclass- Density forthe Chern - Pontryagincharacteristicclassand - Two independentalgebraic invariantsThen the differential invariants:CAN LOCALLY SEE-The TWO HORIZONS-The ERGOSPHERE


gtt =1 - 2M / , where M is the BH massFor gtt = 0.7, 0.0, -0.1, -0.3,-0.5, -1.5, -3.0, - :


Linear perturbations of Kerr metricS. Teukolsky, PRL, 29, 1115 (1972):Separation of thevariables:A trivial dependence on the Killing directions - .(!) :From stability reasons one MUST have:


1972 Teukolsky master equations (TME):The angular equation:The radial equation:Spin:S=-2,-1,0,1,2.andare twoindependentparameters


Up to now only numerical results andapproximate methods were studiedFirst results:• S. Teukolsky, PRL, 29, 1115 (1972).• W Press, S. Teukolsky, AJ 185, 649 (1973).• E. Fackerell, R. Grossman, JMP, 18, 1850 (1977).• E. W. Leaver, Proc. R. Soc. Lond. A 402, 285, (1985).• E. Seidel, CQG, 6, 1057 (1989).For more recent results see, for example:• H. Onozawa, gr-qc/9610048.• E. Berti, V. Cardoso, gr-qc/0401052.and the references therein.


The regularity of the solutions simultaneouslyat the both singular ends of the interval [0,Pi] is:W [ , ] = 0,W – THE WRONSKIAN , or explicitly:It yields the relation:whith unfortunately explicitly unknown function .


Explicit form of the radial Teukolsky equationwhere we are using the standard• Note the symmetry between and in the radial TME• and are regular syngular points of the radial TME• is an irregular singular point of the radial TME


Two independent exact solutions of the radialTeukolsky equation in outer domain are:


BH boundary conditionsat the event horizon:The waves can go only into the horizon.Consequence:- only the solutionobeys BH BC at the EH.If- only the solutionobeys BH BC at the EH.=> An additional physical clarification.


Boundary conditions at spaceinfinity – only going to waves:If, then:If, then:


As a result one has to solve the system ofequations for and : ( )1) For any :2) and when :or=> a nontrivial numerical problem.


Making useof indirectmethods:H. Onozawa,1996


The Relativistic Jets:The Most Powerful andMisterious Phenomenonin the Universe, which areobserved at different scales:1. Around single neutron star (~10-1000 AU)2. In binary BH–Star, and Star-Star systems3. In Gamma Ray Burst (GRB) (~1 kPs)4. Around galactic nuclei (~1 MPs)5. Around galactic collisions (~10 MPs)6. Around galactic clusters (~200 Mps)=> UNIVERSAL NATURE ???


Jets from GRB


A hyper nova 08.09.05 (distance 11.7 bills lys)Formation of WHAT ???: BH???, OR ???VU6APFLG.movSeries of explosions observed!


The Jetfrom M872006 NewsJets fromGRB060418andGRB060607A:~ 200 Earthmasses withvelocity0.999997 c


3C321 Jet :Black Hole Fires at Neighboring Galaxy


Other observed jets:


Today’s theoretical modelsRelativisticJetMolecularTorusMassiveBlack HoleCommon feature:Rotating (Strong)Gravitational FieldAccretionDisk


Another Model – accretion ofmaterial from companion star


Singular solutions of the angularTeukolsky equationBesides regular solutions the angular TME has singular solutions:and


The singularities can be essentially weakened ifone works with Polynomial Heun’s functions(analogy with Hydrogen atom):Three termsrecurrence relation:Polynomial solutions with:andDefines symple functions


Examples of Relativistic Jets 1


Examples of Relativistic Jets 2


Some animations of our jet model


Double wafes (with different velocities):amplitude wave and phase waveRegular solution of angular TME with three nodes:The phase wave:The amplitude wave:


Double wafes (with different velocities):amplitude wave and phase waveJet solutions of the angular TMEThe phase wave:The amplitude wave:


The distribution of the eigenvalues in the complexplane for the singular case s=-2, m=1 withF(z)=zF(z)=1/z


The singular case s=-2, m=1 with, 2M=1, a/M=0.99Re(omega) Im(omega)0.17288 -0.009440.18630 -0.055640.22508 -0.076920.30106 -0.090090.33533 -0.098810.38281 -0.099090.35075 -0.120080.27110 -0.130290.47609 -0.152000.47601 -0.160000.60080 -0.180230.56077 -0.250760.50049 -0.299450.40205 -0.37716


Problems in progress: Imposing BH boundary conditions one canobtain and improve the known numerical results=> a more systematic of the QNM in outerdomain. QNM of the Kerr metric in the BH interior. Novel models of the central engine of GRB Imposing Dirichlet boundary conditions one canobtain new models of rotating compact objects. More systematic study of QNM of neutron stars. Study of the still unknown QNM of gravastars.


Physon:The pink clusterhttp://physon.phys.uni-sofia.bg/IndexPageAt present:32 processorsPerformance:Up to128 GFlops


Some basic conclusions:• Heun’s functions are a powerful tool for study of all types ofsolutions of the Regge-Wheer and the Teukolsky masterequations.• Using Heun’s functions one can easily study differentboundary problems for perturbations of metric.• The solution of the Dirichlet boundary problem gives anunique hint for the experimental study of the old problem:Whether in the observed in the Natureinvisible very compact objects with strong gravitational fieldsthere exist really hole in the space-time ?=> resolution of the problem of the real existence of BH• The exact singular solutions of TME can describe relativisticjets.


Thank You

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